The purpose of this study is to develop thymic transplantation as a therapy for complete DiGeorge syndrome. The patient population includes only those patients with DiGeorge syndrome who have no T cells and no significant T cell function upon repeated in vitro testing of peripheral blood. Human postnatal thymic tissue is obtained after informed consent from infants undergoing heart surgery. (This thymic tissue normally would be discarded.) The thymic tissue is cultured in the laboratory for two weeks prior to transplant to allow time for safety testing of the donor (for EBV, CMV, hepatitis B and C, and HIV 1 and 2) and to allow mature T cells in the donor thymus to die. The thymus is transplanted into the quadriceps muscle of the recipient. The patient is followed after transplantation by periodic assessment of T cell numbers and function. T cell function is assessed by response of peripheral blood mononuclear cells to mitogens, by mixed lymphocyte reactions, and by in vitro responses after immunization with tetanus toxoid. A biopsy of the transplanted thymus is done at three months after the transplant. Monoclonal antibody analysis of frozen sections of the biopsy are used to determine whether there is host thymopoiesis occurring in the donor epithelium. HLA antibodies distinguish donor from host cells.